# APPENDIX B: PEAK-TO-PEAK ANALYSIS

Peak-to-peak analysis is a relatively simple method to assess the capacity utilization of an industry over time. An advantage of peak-to-peak analysis is that it requires information on only one output measure and one input measure, and hence is suited to estimating capacity utilization with only Level 1 data (see Table A.1). Peak-to-peak analysis has been applied in fisheries by Ballard and Roberts (1977), Ballard and Blomo (1978) and Hsu (2003).

## The underlying theory

Peak-to-peak analysis is based on an underlying assumption that output is a function of the level of inputs and a technology trend, such that

(1)

where Yt is the output in time, t; a0 is a proportionality constant; Vt is a composite or aggregate index of inputs; and Tt is the technology trend that represents productivity change. An implicit assumption in the use of a composite index of inputs is that the technology displays constant returns to scale. That is, increasing all inputs will result in a proportional increase in output.

The level of technology is determined by the average rate of change in productivity between peak years, where productivity is given by Yt/Vt (i.e. average output per unit of input). The technology in any one year is thus

(2)

where m is the length of time from the previous peak year, and n is the length of time to the following peak year, and Tt-m is the level of technology at the previous peak (i.e. year m) equivalent to the average productivity (e.g. catch per unit of effort) in that period. The other term on the right hand side (i.e. the term inside the brackets) represents the cumulative change in productivity between the two peaks. This is added to the average productivity in the previous peak year (i.e. year m) to give an estimate of the average productivity of capacity in subsequent years.

An alternative way of estimating the level of technology between peaks is given by

(3)

where Yn/Vn is the average productivity in the upper peak and Ym/Vm is the average productivity in the lower peak. The term in the brackets represents the average change in productivity between the two peaks. Both approaches produce identical results.

Assuming the proportionality constant has a value of 1, the estimate of the level of technology is equivalent to the capacity level of productivity (i.e. Tt = Yt*/Vt, where Yt* is the capacity level of output). From this, the capacity level of production can be estimated from the product of the inputs and the capacity level of productivity, such that

Yt* = VtTt (4)

and capacity utilization can be estimated by

CUt = Yt*/Yt. (5)

A particular difficulty in interpreting the results of a peak-to-peak analysis in fisheries is that no consideration is given to changes in the stock level. Apparent changes in productivity may be due to either changes in technology (the underlying assumption of the technique) or changes in the stock level.

This problem may be particularly pertinent in developing fisheries, where catch rates may increase rapidly initially, with the main peak occurring in the middle of the time series. Subsequent declines in catch rates may reflect falling stock levels. However, if the main peak is used as the last peak in the series (all other years showing a steady decline), it is likely that the technique will over-estimate capacity output and under-estimate capacity utilization.

The problem can be minimized by including lower peaks rather than successively higher peaks as is generally used in other industries that do not rely upon a biological resource base.

## Example of use: Nigerian artisanal fishing sector

Data on the artisanal fishing sector in Nigeria were used as an example of how peak-to-peak analysis can be used to estimate capacity. The data were derived from Amire (2003), and are presented in Table B.1.

Table B.1 - Nigerian artisanal fisheries productivity, 1976-1994

 Average catch per: Year Canoes Fishers Production Canoe Fisher 1976 134 337 413 832 327 561 2 438 0.792 1977 137 447 424 838 331 280 2 410 0.780 1978 138 447 425 298 336 138 2 431 0.790 1979 133 728 446 152 356 888 2 669 0.800 1980 133 723 459 065 274 158 2 050 0.597 1981 120 142 440 592 323 916 2 696 0.735 1982 105 239 416 959 377 683 3 589 0.906 1983 129 555 472 122 376 984 2 910 0.798 1984 109 638 342 219 246 784 2 251 0.721 1985 80 688 302 234 140 873 1 746 0.466 1986 77 134 408 927 160 169 2 077 0.392 1987 76 644 437 465 145 755 1 902 0.333 1988 77 144 447 850 185 181 2 400 0.413 1989 77 155 470 250 171 332 2 221 0.364 1990 76 981 452 187 170 459 2 214 0.377 1991 77 093 457 102 168 211 2.182 0.368 1992 77 076 459 847 184 407 2 393 0.401 1993 77 050 456 381 106 276 1 379 0.233 1994 77 073 457 775 124 117 1 610 0.271

Source: Amire (2003).

The choice of input may have an impact on the measure of capacity output and, consequently, capacity utilization. In the Nigerian artisanal fleet, the number of canoes active in the fishery had declined over time while the number of fishers remained relatively constant (a result of more fishers operating per canoe). Over the same period, motorization increased in the fishery from 8.7 percent in 1996 to 20.8 percent in 1994 (Amire, 2003). As a result, it would be expected that there was substantial technological change in the fishery. Developing a composite index of inputs in such a case is difficult without first estimating a production function and imposing constant returns to scale.

For purposes of illustration, capacity was assessed using both canoes and fishers (separately) for the input measure. From Table B.1, it can be seen that the peak productivity periods for both inputs were 1976, 1979, 1982, 1988 and 1992. These peaks also are apparent by graphing the catch per unit input series (Figure B.1).

Figure B.1 - Catch-per-unit input, Nigerian artisanal fleet

Table B.2 - Peak-to-peak analysis using canoes as input measure

 Year (t) Canoes (Vt) Production (Yt) CPUE (Yt/Vt) Average technological changea Capacity CPUE (Tt) Capacity output (Yt*) Utilization rate (Yt/Yt*) 1976 134 337 327 561 2 438 - 2 438 327 561 100% 1977 137 447 331 280 2 410 0.0768 2 515 345 701 96% 1978 138 247 336 138 2 431 0.0768 2 592 358 330 94% 1979 133 728 356 888 2 669 0.0768 2 669 356 888 100% 1980 133 723 274 158 2 050 0.3067 2 975 397 885 69% 1981 120 142 323 916 2 696 0.3067 3 282 394 321 82% 1982 105 239 377 683 3 589 0.3067 3 589 377 683 100% 1983 129 555 376 984 2 910 -0.1981 3 391 439 289 86% 1984 109 638 246 784 2 251 -0.1981 3 193 350 041 71% 1985 80 688 140 873 1 746 -0.1981 2 995 241 631 58% 1986 77 134 160 169 2 077 -0.1981 2 797 215 711 15% 1987 76 644 145 755 1 902 -0.1981 2 599 199 161 73% 1988 77 144 185 181 2 400 -0.1981 2 400 185 181 100% 1989 77 155 171 332 2 221 -0.0020 2 398 185 055 93% 1990 76 981 170 459 2 214 -0.0020 2 396 184 485 92% 1991 77 093 168 211 2 182 -0.0020 2 395 184 600 91% 1992 77 076 184 407 2 393 -0.0020 2 393 184 407 100% 1993 77 050 106 276 1 379 -0.0020 2 391 184 192 58% 1994 77 073 124 117 1 610 -0.0020 2 389 184 094 67%

Note: Peak years in bold, a) estimated by [(Yn/Vn)-(Ym/Vm)]/(n-m)

The analyses, undertaken in an Excel spreadsheet, are given in Tables B.2 and B.3 using canoes and fisher numbers respectively. Average technological change was estimated between the peak years (indicated in bold). For example, between 1976 and 1979, average productivity change was (2.669-2.438)/(4-1) = 0.0768.

Capacity CPUE is estimated by adding the average technological change to the preceding year’s value. Capacity output is estimated by multiplying the capacity CPUE by the input level. The utilization rate is estimated by dividing actual output by capacity output.

Table B.3 - Peak-to-peak analysis using number of fishers as input measure

 Year (t) Fishers (Vt) Production (Yt) CPUE (Yt/Vt) Average technological changea Capacity CPUE (Tt) Capacity output (Yt*) Utilization rate (Yt/Yt*) 1976 413 832 327 561 0.792 0.792 327 561 100% 1977 424 838 33 1280 0.780 0.003 0.794 337 461 98% 1978 425 298 336 138 0.790 0.003 0.797 339 016 99% 1979 446 152 356 888 0.800 0.003 0.800 356 888 100% 1980 459 065 274 158 0.597 0.035 0.835 383 419 72% 1981 440 592 323 916 0.735 0.035 0.871 383 540 84% 1982 416 959 377 683 0.906 0.035 0.906 377 683 100% 1983 472 122 376 984 0.798 -0.082 0.824 388 911 97% 1984 342 219 246 784 0.721 -0.082 0.742 253 823 97% 1985 302 234 140 873 0.466 -0.082 0.660 199 368 71% 1986 408 927 160 169 0.392 -0.082 0.578 236 194 68% 1987 437 465 145 755 0.333 -0.082 0.496 216 782 67% 1988 447 850 185 181 0.413 -0.082 0.413 185 181 100% 1989 470 250 171 332 0.364 -0.003 0.410 192 977 89% 1990 452 187 170 459 0.377 -0.003 0.407 184 155 93% 1991 457 102 168 211 0.368 -0.003 0.404 184 731 91% 1992 459 847 184 407 0.401 -0.003 0.401 184 407 100% 1993 456 381 106 276 0.233 -0.003 0.398 181 594 59% 1994 457 775 124 117 0.271 -0.003 0.395 180 722 69%

Note: Peak years in bold, a) estimated by [(Yn/Vn)-(Ym/Vm)]/(n-m)

Despite differences in the input measure used, the estimated capacity output was fairly similar in both instances (Figure B.2a). The estimated capacity utilization in each year was also relatively similar (Figure B.2b).

Figure B.2 -a) estimated capacity and b) estimated capacity utilization